On Probabilities for Extreme Values of Sums of Random Variables Defined on a Homogeneous Markov Chain with a Finite Number of States

This paper examines the probabilities for values of sums of random variables defined on a homogeneous Markov chain with a finite number of states. These values are such that their deviations from the smallest or largest possible value for each instant of time "$n$" are bounded in their sum. By separating traj ectorits in the random walk into classes defined by a proper method, regular components are picked out from the probabilities under consideration and exact and asymptotic formulas are found (for $n\to\infty$) for each of these components.